Slope fields are a very important tool in analyzing, understanding and using many differential equations most notably non-linear ones: it's one thing to solve it, quite another to intutively understand what's going on. Why does my removal of one measly fish out of my grandpa's pond cause the entire fish population to collapse? You need to get Blanchard, Devaney, and Hall, "Differential Equations". Spend some time with it, six weeks, and you'll be tops telling us what to do. Mathematica has some nice tools for creating phase portraits (slope fields) and especially ver 7 has what, a really nice function I don't remember the name for generating very nice pictures of slope fields. Start with some simple ones: coupled set of two linear equations and generate the slope field, then go to more complicated ones.